Finite Group Databasebeta

← #6532185

Subgroups — #6532185

Order 512.

1608 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2546C211yesmaximal, characteristic
2#34187C211yesmaximal, characteristic
3#27616C214yesmaximal
4#27613C212yesmaximal
5#27610C212yesmaximal
6#26294C218yesmaximal, complemented
7#26292C218yesmaximal, complemented
8#21840C212yesmaximal
9#21822C211yesmaximal, characteristic
10#21363C212yesmaximal
11C22×C4⋊Q8—21no—
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C2×#432C2212yes—
15#2546—22no—
16#2546—22no—
17#2546C2214yes—
18#2546C2214yes—
19C2×#398—21no—
20C2×#398—21no—
21#2356—22no—
22#2356—22no—
23#2355C2214yes—
24#2355C2214yes—
25#2313—21no—
26#2313—21no—
27#2310—21no—
28C2×#391—21no—
29C2×#389C2212yes—
30C2×#389C2212yes—
31#2285C2211yescharacteristic
32#2285C2211yescharacteristic
33#2283C2212yes—
34#2283C2212yes—
35#2274—21no—
36C2×C4×C4⋊C4C2211yescharacteristic
37#2101C2218yes—
38#2100C2214yes—
39#2099C2214yes—
40#2085C2218yes—
41#2084C2214yes—
42#2083C2214yes—
43#1993C2218yes—
44#1987C2218yes—
45#1985C2214yes—
46#1985C2214yes—
47#1949C2218yes—
48#1921C2218yes—
49#1920C2218yes—
50#1919C2218yes—
51#1918C2218yes—
52#1853C2214yes—
53#1852C2214yes—
54#1823C2214yes—
55#1823C2214yes—
56#1817C2212yes—
57#1817C2212yes—
58#1776C2218yes—
59C23×Q8—21no—
60C23×Q8D411yescharacteristic
61C2×C4⋊Q8—24no—
62C2×C4⋊Q8—24no—
63C2×C4⋊Q8—22no—
64C2×C4⋊Q8—28no—
65C2×C4⋊Q8—48no—
66C2×C4⋊Q8C2314yes—
67C2×C4×Q8—28no—
68C2×C4×Q8D412yes—
69C2×C4×Q8D412yes—
70C2×C4×Q8C2312yes—
71C2×C4×Q8C2312yes—
72C2×#117—21no—
73C2×#117—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C42—21no—
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86#432—28no—
87C2×#107C2318yes—
88C2×#106C2318yes—
89C2×C4⋊C8C2314yes—
90C2×C22⋊C8C2314yes—
91C2×#97C2314yes—
92#398—24no—
93#398—24no—
94#398—24no—
95#398—24no—
96#391—24no—
97#391—24no—
98#391—24no—
99#391—24no—
100#389—24no—
101#389—22no—
102#389—22no—
103#389C2314yes—
104#389C2314yes—
105#389C2318yes—
106#387—24no—
107#387—24no—
108#387—24no—
109#387—24no—
110#384C2312yes—
111#384C2312yes—
112#383—22no—
113#383—22no—
114#382—21no—
115#382—21no—
116#382C2314yes—
117#382C2314yes—
118#381—22no—
119#381—22no—
120#380—21no—
121#380—21no—
122#380C2312yes—
123#380C2312yes—
124C4×C4⋊C4—24no—
125C4×C4⋊C4—24no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4C2312yes—
130C4×C4⋊C4C2312yes—
131C4×C4⋊C4C2314yes—
132C4×C4⋊C4C2314yes—
133C4×C4⋊C4C2314yes—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—22no—
136#376—22no—
137#376—22no—
138C2×#95—21no—
139C2×#95—22no—
140C2×#95—22no—
141#340C2318yes—
142#340C2318yes—
143#340C2318yes—
144#340C2318yes—
145#340C2318yes—
146#340C2318yes—
147#336C2314yes—
148#336C2314yes—
149#336C2318yes—
150C22×Q8—24no—
151C22×Q8—24no—
152C22×Q8—24no—
153C22×Q8—42no—
154C22×Q8—44no—
155C22×Q8—44no—
156C22×Q8—44no—
157C22×Q8C2×D412yes—
158C22×Q8C2×D412yes—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2411yescharacteristic
168C22×C8C2418yes—
169C4⋊Q8—28no—
170C4⋊Q8—28no—
171C4⋊Q8—416no—
172C4⋊Q8—416no—
173C4⋊Q8—48no—
174C4×Q8—28no—
175C4×Q8—28no—
176C4×Q8—24no—
177C4×Q8—24no—
178C4×Q8—24no—
179C4×Q8—24no—
180#117—44no—
181#117—44no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—48no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—48no—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2414yes—
207C2×C4⋊C4C2414yes—
208C2×C4⋊C4C2414yes—
209C2×C4⋊C4C2414yes—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C42—24no—
213C2×C42—24no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—24no—
217C2×C42—24no—
218C2×C42—24no—
219C2×C42—44no—
220C2×C42C2×D412yes—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D412yes—
224C2×C42#4118yes—
225C2×C42C2412yes—
226C2×C42C2411yescharacteristic
227C2×C42C2411yescharacteristic
228C2×C42C2412yes—
229#107—216no—
230#106—216no—
231C4⋊C8—28no—
232C22⋊C8—28no—
233#97—28no—
234#95—24no—
235#95—28no—
236#95—24no—
237#95—24no—
238#95—22no—
239#95—22no—
240#95—24no—
241#95—24no—
242#95—24no—
243#95—24no—
244#95—22no—
245#95—22no—
246#95—24no—
247#95—24no—
248C24C22×D411yescharacteristic
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—24no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—416no—
256C2×Q8—48no—
257C2×Q8—48no—
258C2×Q8—48no—
259C2×Q8—816no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—28no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4#12314yes—
284C22×C4#12314yes—
285C22×C4AGL(1,ℤ/8)14yes—
286C22×C4C22×D412yes—
287C22×C4C22×D411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D412yes—
290C22×C4C22×D412yes—
291C22×C4C22×D412yes—
292C22×C4C22×D411yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C2×#4114yes—
296C22×C4C2511yescharacteristic
297C2×C8—28no—
298C2×C8—28no—
299C2×C8—28no—
300C4⋊C4—28no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—28no—
309C4⋊C4—28no—
310C4⋊C4—28no—
311C4⋊C4—416no—
312C4⋊C4—44no—
313C4⋊C4—44no—
314C4⋊C4—48no—
315C4⋊C4—48no—
316C4⋊C4—48no—
317C4⋊C4—44no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C42−1+418yes—
321C22⋊C4—44no—
322C22⋊C4—44no—
323C42—22no—
324C42—22no—
325C42—24no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—48no—
331C42—44no—
332C42—44no—
333C42—84no—
334C42—84no—
335C42#12318yes—
336C42C2×#4114yes—
337C42C2×#4114yes—
338C23—22no—
339C23—22no—
340C23C2×#12312yes—
341C23C2×AGL(1,ℤ/8)12yes—
342C23#57512yes—
343C23C23×D411yescharacteristic
344Q8—48no—
345Q8—48no—
346Q8—816no—
347C2×C4—24no—
348C2×C4—28no—
349C2×C4—28no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—28no—
358C2×C4—24no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—28no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—44no—
369C2×C4—416no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—88no—
385C2×C4C2×#12312yes—
386C2×C4C2×#12314yes—
387C2×C4C2×#12314yes—
388C2×C4C2×AGL(1,ℤ/8)12yes—
389C2×C4C2×AGL(1,ℤ/8)12yes—
390C2×C4C22×#4112yes—
391C2×C4C2×2−1+414yes—
392C8—48no—
393C22—22no—
394C22—22no—
395C22—24no—
396C22—24no—
397C22#309414yes—
398C22C22×#12311yescharacteristic
399C22#348612yes—
400C22C22×AGL(1,ℤ/8)11yescharacteristic
401C22C2×#57511yescharacteristic
402C22#358612yes—
403C4—22no—
404C4—24no—
405C4—22no—
406C4—22no—
407C4—24no—
408C4—42no—
409C4—42no—
410C4—41no—
411C4—41no—
412C4—48no—
413C4—84no—
414C4—84no—
415C4—84no—
416C4#350914yes—
417C2—24no—
418C2#3567411yescharacteristic
419C2C2×#309411yescharacteristic
420C2#6051911yescharacteristic
421C2#6182712yes—
422C2#6183212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
6. #26294C2semidirect
7. #26292C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#25462
#341872
#276162
#276132
#276102
#262942
#262922
#218402
#218222
#213632